Suppose I play 1. e4 and my opponent plays 1..e5. Do FIDE tournament rules require me to write 1...e5 on my scoresheet before I play the next move 2. Nf3?
No, you don't have to. In rule 8.1 of the Laws of Chess, it says:
A player may reply to his opponent’s move before recording it, if he so wishes. He must record his previous move before making another.
So your previous move must have been recorded by the time you make the next. You must have recorded at least 1.e4 before you can play 2.Nf3.